Since people tend to know some practical applications of this mitsvah better than the mitsvah itself, we will start with those and work our way back to a general understanding. In the 5^th chapter of Yesodei haTorah, Rambam assumes this mitsvah is the reason for yehareg ve-`al ya`avor, for a Jew’s being obligated to allow him/herself to be killed rather than to transgress a sin in certain situations. To understand this mitsvah, then, a review of yehareg ve`al ya`avor is in order.

There are three sins a Jew must never commit, even under threat of death: idolatry, murder, and gilui `arayot (sexual relationships prohibited to such an extent that kiddushin would have no halakhic validity). That rule applies regardless of who would see the sin, the reason the person coercing the Jew wants him to sin, and the social/political context where the threat is made, factors that will affect other applications of yehareg ve-`al ya`avor. Thinking about why these three are so important might start us on the road to understanding the broader mitsvah as well.

It seems simple to connect the obligation to die rather than commit idolatry to an obligation to sanctify God’s Name, since in worshipping a foreign god we are most obviously rejecting the notion of a single God. Refusing to do so, even on pain of death clearly shows our devotion to the sanctity of His Great Name, the simple sense of the mitsvah.

Murder fits less obviously into this mitsvah. The gemara says that it is simply logical, since the coerced murderer cannot know that his/her blood is "redder" than the victim's. That logic explains why it is prohibited to kill someone else even at the cost of one’s own life, but does not show how that serves as an example of kiddush Hashem. The Sefer haHinukh’s phrasing of the gemara’s logic shows how he combined the two. He says that we cannot know which of the two-- the potential murderer or the victim-- will, in the time left after this incident, perform more mitsvot. In that way of reading it, the gemara's comment about not knowing whose blood is redder really means that we do not know who will perform more mitsvot. If so, the definition of "redder" blood is a person’s future potential as a worshipper of God, something humans cannot ever know.

Noting that the question of who should live or die depends on who will serve God better in the future, along with recognizing that we cannot judge that question, explains how our refusal to murder someone else is a kiddush Hashem. We are not refusing to murder the person simply because it is unethical to kill (although it certainly is unethical to kill). Rather, we are recognizing that the standard of judgement of the value of human life is beyond our capabilities as human beings; as we get killed for refusing to murder someone else, it is inherently an act of submission not only to mitsvot but to the unbridgeable gap between our judgement and God’s. The derivation of gilui arayot could follow similar lines, but requires much more detailed discussion than is possible here, so we will leave it for another time.

We are also sometimes required to allow ourselves to be killed rather than commit other sins. In those cases, it is the circumstances rather than the sin itself that lead to that requirement. If, for example, a non-Jew would order a Jew publicly, under threat of death, to wear shatnes (note: publicly here means in front of ten Jews; some authorities assume that women would count for this minyan, since they are equally obligated in issues of kiddush Hashem), the Jew would have to refuse. Alternatively, if a government were making a concerted effort to stamp out Jewish observance, Jews would be required to be killed even if the order to transgress (and the threat of death) was issued in private. According to many authorities, however, if the non-Jew was simply trying to get us to do something that he personally wanted (sew him a suit on Shabbat, e.g.), we would be allowed (and maybe even required) to do so even in public or in times of shemad.

These cases suggest that kiddush Hashem, as Rambam sees it, means putting God's goals in the world ahead of our own, particularly when specifically confronted with a choice between what God wants and our own lives. In that sense, the Big Three would be acts that inherently contravene central elements of God's picture of the world, so that even private violations count as saving our lives at the cost of violating God's plan. For other mitsvot, the acts themselves are not so central to God’s rule of the world that we must forego our lives in order to avoid transgressing them. When, however, those acts are made into crucial choices -- in times when the whole religion is under attack or when our choice is being made in public (so that other Jews will see us choosing life over God)--we are obligated to choose God.

That kiddush Hashem involves choosing God over life in extreme examples explains another version of kiddush Hashem offered by the Sefer haHinukh. According to the Hinukh, someone who wantonly sins in private (with no coercion and with clear recognition that it is a sin), desecrates God’s Name; conversely, someone who refrains from sin simply because God prohibited it, even when acting completely alone, sanctifies the Name. Here, too, it is the choice of God’s Will over ours, the submission to His judgement of right and wrong, good and evil, the constitutes a sanctification of His Name. Shabbat Shalom.